2011 HiMCM B题特等奖学生论文下载3074

2011 HiMCM B题特等奖学生论文下载3074

 

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论文摘要如下：

14th Annual High School Mathematical Contest in Modeling (HiMCM) Summary Sheet

Team Control Number: 3074 Problem Chosen: B

I have lost my ring! We all lose things, and the hardest part is always finding where we left them. In Problem B we were assigned the intimidating task of finding an object in a large park, and a lost jogger in an even larger park using only a penlight. Needless to say, without the mathematical and computer models contained herein, it would be like finding a needle in a haystack. However, this task is not the impossibility that it may seem at first.

When searching for a lost object, it is important to first determine the locations where there is the greatest probability of finding it. Once this has been accomplished, the next step is to develop a method for searching those locations efficiently. In both problems, we are asked to search for an object with few initial clues as to its location. However, we can make several assumptions about how the objects, and those searching for them, behave. A lost ring, for example, will not arbitrarily change its location. We therefore base our method of search for the ring off of our subject, the person who lost the ring, who is far more likely to have changed his location, and whose location changes may be predictable. To find where the ring is located, we must focus our model on where our subject would have spent the most time because he could have only dropped the ring in a place where he was present. If we are able to determine the areas where our subject spent the majority of his time, or was likely to have spent the majority of his time, those areas will be where we are most likely to find the ring. In order to determine where these areas of high probability are, we need to answer two questions: First, where must our subject have gone? And second, where could he have remained for a period of time? Because the park can only be accessed by car, we can therefore deduce that Tim must have parked his car at one of the fifteen parking lots. The areas near parking lots, therefore, are much more likely to have been places that Tim walked than other stretches of like Aikens road. In addition, Tim is likely to have stopped for a period of time at picnic tables, restrooms, or the Swimming Pond. If he stopped at these areas, the area of trail surrounding them is also likely to have been covered by our subject walking to or from the location. We have now established where the ring is most likely to be found, it is most likely found at or around one of the locations mentioned previously: parking lots, restrooms, and picnic tables.

Having determined the criteria necessary to determine whether an area is beneficial to search, being close to the attractions of the park, we can construct a probability density map that for any point in the park will determine how likely it is that the ring is located at that point. Using this map, we must construct a path that allows us to search all of the highest density, highest probability, areas as quickly as possible. We start at each high-probability location, and attempt to find the shortest route from each location that passes through all the others. We then select the shortest of these routes, which is about 3.9 miles. As Tim can walk 8 miles in the allotted 2 hours, this leaves him with 5.1 miles worth of walking to pass by as many other facilities as possible and return to the starting point. This will enable him to cover the 80\% of the road that gives him the highest probability of finding the ring.

In the second scenario, the search for Sam, our subject, presents a different problem: how to find an object that may or may not be moving. Because we cannot assume anything about Sam’s behavior as we search for him (we assume that he will not go in closed-off areas), we are forced to gain our information from his natural inclinations, previous behavior studies, and from the park’s layout.

As in the first problem, we can assume that Sam must have accessed the park by car. Therefore, there are five possible places where his run could have begun. We also assume that, as he planned to run for five miles, so at most he would have intended to turn around after 2.5 miles and then turn around. It is therefore safe to assume that at the time Sam became lost, he was within a 2.5-mile radius of one of the parking lots. He therefore is most likely to be found in an area that is within 2.5 miles of multiple parking lots. We may also assume that Sam is not lost if he is in a parking lot or at another recognizable landmark. He is therefore is most likely to be found at the edge of this 2.5-mile range, where he is farthest away from these landmark.

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